A universal constant for exponential Riesz
sequences
A. M. Lindner∗
Abstract
The aim of this paper is to study certain correlations between lower
and upper bounds of exponential Riesz sequences, in particular between
the sharp lower and upper bounds, where we show that the product of the
sharp bounds of an exponential Riesz sequence is bounded from above by
a universal constant. The result is applied to the norms of coefficient and
frame operators and their inverses.
Keywords: exponential Riesz sequence, exponential Riesz basis, exponential
frame, sharp bounds
AMS subject classification: 42 C 15, 94 A 12
1
Introduction
Let H be a separable Hilbert space over C. A sequence Φ = (ϕn )n∈Z of elements
in H is called a Riesz-Fischer-sequence, resp. a Bessel sequence for H, if there
is A > 0, resp. B > 0, such that for all natural numbers n and all complex
numbers c−n , . . . , cn :
2
X
n
c
ϕ
|cj |2 ≤ A·
j j ,
j=−n
j=−n
n
X
(1)
H
2
X
n
X
n
|cj |2 ,
cj ϕj ≤
B
·
resp. j=−n
j=−n
(2)
H
holds. If Φ is both a Bessel and a Riesz-Fischer-sequence, it is called a Riesz
sequence for H. The constants A and B are called lower and upper bounds, respectively. It is an easy matter to check that the supremum of all lower bounds,
resp. the infimum of all upper bounds, of a Riesz sequence is again a lower
bound, resp. an upper bound, which we denote by Aopt (Φ), resp. Bopt (Φ). The
∗ Alexander M. Lindner, Zentrum Mathematik, Technische Universität München, D-80290
München, Germany; e-mail:[email protected]
1
constants Aopt (Φ) and Bopt (Φ) are called the sharp lower and the sharp upper
bounds for Φ.
In this paper, we shall be concerned with exponential Riesz sequences for
L2 (−σ, σ), i.e. with Riesz sequences for L2 (−σ, σ) of the form (eiλn • )n∈Z , where
σ > 0 and (λn )n∈Z is a sequence of complex numbers. From (1) and (2) it follows
readily that if (eiλn • )n∈Z is a Riesz sequence for L2 (−σ, σ), then the sequence
(=λn )n∈Z of imaginary parts must be uniformly bounded.
Young [9, Prop. 1, Cor. 1], [10, Ch. 4, Props. 3 + 2, Th. 3; Ch. 2, Rem.
following Th. 17] has shown that for a sequence (eiλn • )n∈Z of exponentials to be
a Riesz sequence for L2 (−σ, σ) it is sufficient that it be a Riesz-Fischer-sequence
for L2 (−σ, σ), provided (=λn )n∈Z is uniformly bounded. We shall have a closer
look at this result and the occuring bounds. In particular, if σ, τ > 0 and
(λn )n∈Z is a sequence of complex numbers such that
sup |=λn | ≤ τ,
(3)
n∈Z
and (eiλn • )n∈Z is a Riesz-Fischer-sequence for L2 (−σ, σ) with lower bound A,
we shall construct an upper bound B for (eiλn • )n∈Z , depending only on A, σ
and τ (Proposition 1). From this we shall obtain a universal constant for the
product of the sharp bounds of exponential Riesz sequences (Theorem 1).
The results of this paper are part of the author’s doctoral thesis [5, Ch. 4].
2
Results
We need the following
Definition 1. A sequence (λn )n∈Z of complex numers is called separated by
δ > 0, if inf n6=m |λn − λm | ≥ δ. The sequence is called separated, if there is some
δ > 0 such that (λn )n∈Z is separated by δ.
Proposition 1. Let τ ≥ 0, σ > 0, A > 0 and (λn )n∈Z be a sequence of
complex numbers satisfying (3), such that (eiλn • )n∈Z is a Riesz-Fischer-sequence
for L2 (−σ, σ) with lower bound A. Then holds:
1. (λn )n∈Z is separated by
δ := δ(A, σ, τ ) :=
p
1
log (1 + e−στ A/σ).
σ
2. (eiλn • )n∈Z is a Riesz sequence for L2 (−σ, σ) with upper bound B, where
2
2
2
2σ(τ +1)
B := B(A, σ, τ ) := · (e
− 1) · 1 +
=
σ
δ
2
2
· (e2σ(τ +1) − 1) ·
σ
1+
!2
2σ
log (1 + e−στ
p
A/σ)
.
(4)
Remark 1. Proposition 1 states more explicitly a result of Young [9, Prop.
1, Cor. 1], [10, Ch. 4, Props. 3 + 2, Th. 3; Ch. 2, Rem. following Th. 17],
who proved that, under the assumptions of Proposition 1, the sequence (λn )n∈Z
must be separated and that (eiλn • )n∈Z is a Riesz sequence.
From Proposition 1 we shall obtain:
Theorem 1. For every σ > 0 and every τ ≥ 0, there exists a positive
constant C(σ, τ ), such that the following holds:
If (λn )n∈Z is a sequence of complex numbers, satisfying (3), such that Φ :=
(eiλn • )n∈Z is a Riesz sequence for L2 (−σ, σ), then the product of the sharp
bounds Aopt (Φ) and Bopt (Φ) is bounded from above by C(σ, τ ):
Aopt (Φ) · Bopt (Φ) ≤ C(σ, τ ).
(5)
The constant C(σ, τ ) can be choosen as
C(σ, τ ) := 256 · e4στ +2σ · (σ + 1/8)2 .
(6)
Remark 2. There is no universal constant D(σ, τ ) > 0, such that Aopt (Φ) ·
Bopt (Φ) ≥ D(σ, τ ) for all exponential Riesz sequences Φ = (eiλn • )n∈Z for
L2 (−σ, σ) satisfying (3).
Counterexample: For 0 < ε < 1, we define
n
for n ∈ Z\{0}
λεn :=
1 − ε for n = 0
ε
and Φε := (eiλn • )n∈Z . From the orthonormality of ( √12π ein• )n∈Z in L2 (−π, π)
it follows that, for 0 < ε < 1, Φε is a Riesz sequence for L2 (−π, π), and that
ε
ε
Bopt (Φε ) ≤ 4π. However, from keiλ0 • − eiλ1 • k2L2 (−π,π) → 0, for ε → 0, we conclude Aopt (Φε ) → 0, for ε → 0. This shows Aopt (Φε ) · Bopt (Φε ) → 0, for ε → 0.
Definition 2. A sequence Φ = (ϕn )n∈Z in a separable Hilbert space H
over C is called a frame for H (cf. Duffin–Schaeffer [3, Sect. 3]), if there exist
positive constants A, B, such that for all f ∈ H:
X
A · kf k2H ≤
|(f, ϕn )H |2 ≤ B · kf k2H .
n∈Z
The constants A and B are called lower and upper frame bounds, respectively.
The operators
TΦ
SΦ
: H → l2 (Z), f →
7 ((f, ϕn )H )n∈Z and
X
: H → H, f 7→
(f, ϕn )H ϕn
n∈Z
3
are called the coefficient operator, resp. the frame operator, corresponding to
the frame Φ.
Remark 3. From the definition it follows easily that the coefficient operator, corresponding to a frame, is an injective, bounded linear operator, with
bounded inverse on ist range. Furthermore, it can be shown that SΦ is a welldefined, bijective, bounded linear map ( cf. Duffin–Schaeffer [3, Sect. 3]; the
sum converges in the norm of H).
Definition 3. A frame in a separable Hilbert space over C is called an exact
frame (or a Riesz basis), if it is no longer a frame after any of its elements are
removed.
Remark 4. Every exact frame is a Riesz sequence with the same bounds
(cf. Duffin–Schaeffer [3, Lemma X] and Kölzow [4, Sect. II.1, Cor. 1 to Th. 8]).
As a consequence of Theorem 1, we have
Corollary 1. For the constant C(σ, τ ) of Theorem 1 holds:
If (λn )n∈Z is a sequence of complex numbers, satisfying (3), such that Φ :=
(eiλn • )n∈Z is an exact frame for L2 (−σ, σ), then we have the following inequalities for the norms of the coefficient and frame operator and their inverses:
p
C(σ, τ ) · kTΦ−1 k,
kTΦ k ≤
−1
kSΦ k ≤ C(σ, τ ) · kSΦ
k,
−1
2
kTΦ k ≤ C(σ, τ ) · kSΦ k.
Remark 5. An analogue statement to Theorem 2 does not hold for arbitrary
(non exact) exponential frames.
n
Counterexample: We define the sequence Φm := (ei m • )n∈Z , where m ∈ N. It
can be shown that Φm is a frame for L2 (−π, π), but
−1 −1
kTΦm k2 = kTΦ−1
k−2 = kSΦm k = kSΦ
k = 2πm.
m
m
Definition 4. Denote by P Wσ2 the Paley–Wiener–space, consisting of all
entire functions of exponential type at most σ, whose restriction to R belongs
to L2 (R). The norm on P Wσ2 is the usual L2 (R)-norm.
For a sequence Λ = (λn )n∈Z of complex numbers, the operator IΛ is defined by
IΛ : P Wσ2 → CZ , F 7→ (F (λn ))n∈Z .
Remark 6. If, for Λ = (λn )n∈Z , (eiλn • )n∈Z is an exact frame for L2 (−σ, σ),
then IΛ defines a bijective, bounded linear operator from P Wσ2 onto l2 (Z) (i.e.
Λ is a complete interpolating sequence, cf. Young [10, Ch. 4, Th. 9]).
From Corollary 1 we obtain
4
Corollary 2. For the constant C(σ, τ ) of Theorem 1 holds:
If Λ = (λn )n∈Z is a sequence of complex numbers, satisfying (3), such that
(eiλn • )n∈Z is an exact frame for L2 (−σ, σ), then
p
C(σ, τ ) −1
kIΛ k ≤
kIΛ k.
2π
Consequently, kIΛ−1 k is large if kIΛ k is large.
3
Proofs
We shall need the following
Lemma 1 ([6, Lemma 1]). Let δ, σ > 0, τ ≥ 0, and (λn )n∈Z be a sequence
of complex numbers, separated by δ and satisfying (3). Then, for all functions
F of the Paley-Wiener-space P Wσ2 , the following inequality holds:
X
|F (λn )|2 ≤
n∈Z
2 Z
e2σ(τ +1) − 1
2
· 1+
|F (x)|2 dx.
πσ
δ
R
(7)
It should be noted that, for separated real sequences, the first inequality of
type (7) (with a different constant) was given by Plancherel and Pólya [8, p.
126].
Proof of Proposition 1.
1. (For the following proof that (λn )n∈Z is separated we use use similar arguments as Pavlov did in [7, Th. 1]). Let k, n ∈ Z, where k 6= n. Then we have
from (1):
Z
σ
2A = A · (|1|2 + | − 1|2 ) ≤
|eiλk x − eiλn x |2 dx =
−σ
=
Z
σ
|eiλk x |2 · |1 − ei(λn −λk )x |2 dx ≤ e2στ ·
−σ
Z
σ
|1 − ei(λn −λk )x |2 dx.
(8)
−σ
For x ∈ (−σ, σ), we have
∞
∞
X (i(λ − λ )x)k X
(|λn − λk | · |σ|)k
n
k
i(λn −λk )x
|1 − e
|=
= e|λn −λk |σ − 1.
≤
k!
k!
k=1
k=1
Hence we conclude from (8):
2A ≤ e2στ · 2σ(e|λn −λk |σ − 1)2 .
From this it follows easily that
|λn − λk | ≥
p
1
log (1 + e−στ A/σ) = δ(A, σ, τ ),
σ
5
i.e. (λn )n∈Z is separated by δ = δ(A, σ, τ ).
2. From Lemma 1 and assertion 1 we derive (7) with δ = δ(A, σ, τ ). By the
Paley-Wiener-Theorem, this is equivalent to
X
iλn •
|(f, e
n∈Z
2
2
2 2σ(τ +1)
)| ≤ (e
− 1) · 1 +
· kf k2 ,
σ
δ
2
for all f ∈ L2 (−σ, σ). By a theorem of Boas [2, Th. 1] (cf. Young [10, Ch. 4,
Th. 3]), the latter inequality is equivalent to (eiλn • )n∈Z being a Bessel sequence
with upper bound B = B(A, σ, τ ), as defined by (4). 2
Proof of Theorem 1. Let σ > 0, τ ≥ 0, and (λn )n∈Z be a sequence of
complex numbers, satisfying (3), such that Φ := (eiλn • )n∈Z is a Riesz sequence
with sharp bounds A := Aopt (Φ) and Bopt (Φ). From Proposition 1, we conclude
Bopt (Φ) ≤
2(e2σ(τ +1) − 1)
2σ
p
· (1 +
)2 .
σ
log (1 + e−στ A/σ)
(9)
From
A ≤ keiλn • k2L2 (−σ,σ) ≤ 2σ · e2στ
we derive
1
√ · e−στ
8
r
∀n∈Z
A
1
≤ .
σ
2
(10)
Using log(1 + x) ≥ x/2 for x ∈ [0, 1/2[, we thus obtain
log (1 + e−στ
p
p
p
1
1 1
A/σ) ≥ log (1 + √ e−στ A/σ) ≥ · √ · e−στ A/σ.
2
8
8
Using (9) and (10), we conclude
p
√
2
· (e2σ(τ +1) − 1) · (1 + 4 8 σ eστ σ/A)2 ≤
σ
2 2
2
√ στ p
2 2σ(τ +1)
1
1
1
4στ +2σ
≤ ·(e
−1)·
+ σ · 4 8e
σ/A ≤ 256·e
·
+σ · .
σ
8
8
A
Bopt (Φ) ≤
This shows Aopt (Φ) · Bopt (Φ) ≤ C(σ, τ ), for C(σ, τ ) defined by (6). 2
Proof of Corollary 1. Let Φ := (eiλn • )n∈Z be an exact frame for L2 (−σ, σ).
Since the frame bounds of the exact frame Φ coincide with the bounds of Φ as
a Riesz sequence (Remark 4), we conclude that we have, for all f ∈ L2 (−σ, σ),
X
Aopt (Φ) · kf k2 ≤
|(f, eiλn • )||2 ≤ Bopt (Φ) · kf k2 ,
(11)
n∈Z
6
and that Aopt (Φ) and Bopt (Φ) are the best possible constants in this inequality.
From this we derive
q
1
kTΦ k = Bopt (Φ), kTΦ−1 k = p
.
(12)
Aopt (Φ)
−1
The norms of SΦ and SΦ
can also be expressed by the best constants occuring
in (11). It holds
1
−1
kSΦ k = Bopt (Φ), kSΦ
k=
(13)
Aopt (Φ)
(cf. Benedetto and Walnut [1, Th. 3.2 a]). Thus, the theorem follows from
inequalities (5), (12) and (13). 2
Proof of Corollary 2. This follows from Corollary 1 and the Paley–
Wiener–theorem (cf. Young [10, Ch. 4, Th. 18, p. 100]). 2
4
Open questions
1. For what classes of exact frames (not of exponential type) do analogues
to Theorems 1 and 2 exist? It can be thought, e.g., of the class of all
exact Gabor-frames with the discretisation parameters t0 , ω0 , such that
t0 · ω0 = 1.
2. What is the best constant Copt (σ, τ ), fulfilling inequality (5)?
Conjecture: Copt (π, 0) = 4π 2 .
Acknowlegment: The author would like to thank his thesis advisor D. Kölzow
for his steady support.
References
[1] Benedetto, J. J. and Walnut, D. F. Gabor frames for L2 and related spaces.
In: Wavelets: Mathematics and Applications (Benedetto, J. J. and Frazier,
M. W., eds.), CRC Press, Boca Raton (1994), 97-162.
[2] Boas, J. R., Jr. A general moment problem. Am. J. Math., 63 (1941), 361370.
[3] Duffin, R. J. and Schaeffer, A. C. A class of nonharmonic Fourier series.
Trans. Amer. Math. Soc., 72 (1952), 341-366.
[4] Kölzow, D. Wavelets. A tutorial and a bibliography. Rend. Ist. Matem.
Trieste, 26, Suppl. (1994).
[5] Lindner, A. Über exponentielle Rahmen, insbesondere exponentielle RieszBasen. Dissertation, Erlangen–Nürnberg (1999).
7
[6] Lindner, A. M. On lower bounds of exponential frames. J. Fourier Anal.
Appl., 5 No. 2 (1999), 187-194.
[7] Pavlov, B. S. Basicity of an exponential system and Muckenhoupt’s condition. Dokl. Akad. Nauk SSSR, 247 (1979), 37-40 (russian); english translation in Soviet Math. Dokl., 20 (1979), 655-659.
[8] Plancherel, M. and Pólya, G. Fonctions entières et intégrales de Fourier
multiples (Seconde partie). Comment. Math. Helv., 10 (1938), 110-163.
[9] Young, R. M. Interpolation in a classical Hilbert space of entire functions.
Trans. Am. Math. Soc., 192 (1974), 97-114.
[10] Young, R. M. An introduction to nonharmonic Fourier series. Academic
Press, New York (1980).
8